Now, we deal with the injectivity of sampling operator \(T\)  on \(\mathcal {X}\)  given in (REF ). The conditions for injectivity of the sampling operator for the case of the union of closed subspaces have been passed to those of single space [1]}. But here the problem is sum of two MI spaces \(\mathcal {N}_{\delta , \theta }:=\mathcal {N}_{\delta }+\mathcal {N}_{\theta }=\lbrace u+v: u\in \mathcal {N}_{\delta }, v \in \mathcal {N}_{\theta }\rbrace \)  need not to be closed and therefore \(\mathcal {N}_{\delta ,\theta }\) 
need not to be an MI space. This directly shifts the problem into the field of closedness of subspaces where we will take the help of Section . The following Proposition suggests that the closure of the sum of two MI spaces is again an MI space generated by the union of the generators of two MI spaces and thus for the space \(\mathcal {N}_{\delta ,\theta }\)  to be an MI space, it is enough for it to be closed. A similar result for the case of shift invariant spaces can be found in [2]}.
